Exercise 16, § 5

a) Let $(f_n)$ be a sequence of functions in $\mathscr{L}^1(\mathrm{T}, \mu)$. Show that the following conditions are equivalent:

α) The sequence $(\widetilde{f}_n)$ is convergent in $L^1$ for the weakened topology $\sigma(L^1, L^\infty)$.

β) The set of $\widetilde{f}_n$ is relatively compact in $L^1$ for the weakened topology, and the sequence of measures $f_n \cdot \mu$ converges vaguely in $\mathscr{M}(\mathrm{T})$.

γ) For every open subset $\mathrm{U}$ of $\mathrm{T}$, the sequence of numbers $\int_{\mathrm{U}} f_n \, d\mu$ is convergent in $\mathbf{R}$.

(To prove that β) implies α), use criterion α) of Exer. 15 and the definition of measurable function. To prove that γ) implies β), consider first the special case of the space $L^1(\mathbf{N})$ (for the discrete measure defined by the mass +1 at each point) using Exer. 15 of TVS, IV, §2. In the general case, apply the criterion ζ) of Exer. 15 to reduce to the case of $L^1(\mathbf{N})$, by associating to each $f_n$ the summable sequence $(\int_{\mathrm{U}_m} f_n \, d\mu)_{m \in \mathbf{N}}$.)

b) Deduce from these results that, in $L^1$, every Cauchy sequence for the weakened topology is convergent for that topology.

c) On the interval $\mathrm{T} = [0, 1]$, let $\mu$ be Lebesgue measure and, for every integer $n \geqslant 1$, let $\mu_n$ be the measure defined by the mass $1/n$ at each of the points $k/n$ ($0 \leqslant k < n$); let $\nu$ be a positive measure on $\mathrm{T}$ such that $\mu = g \cdot \nu$, $\mu_n = f_n \cdot \nu$ (Exer. 4 b)). Show that the sequence $(\mu_n)$ converges vaguely to $\mu$, $\mu_n(A)$ tends to $\mu(A)$ for every finite subset $A$ of $\mathrm{T}$ and for every open set $A$ such that the measure induced by $\nu$ on the boundary of $A$ has finite support, but that $\mu_n(U)$ does not tend to $\mu(U)$ for all of the open sets $U \subset \mathrm{T}$ (cf. Exer. 13 b)).